Past exam of the further mathematical biology course of the University of Oxford / 2023 / 1 / b / Solution

The unit integrals of the profiles make $N_i(t)={\int }_0^{\infty }{n}_i(\tau ,t)d\tau$ the phase populations, and $f_i$ their normalized cell-cycle age densities. Assume the age tails vanish and the following integrals are finite. For a fixed profile define its mean exit hazard function byThese constants have units of inverse time. Integrating the age-structured population equations and using the boundary conditions givesThe incoming fluxes are $n_1(0,t)=2{\Lambda }_2{N}_2$ and $n_2(0,t)={\Lambda }_1{N}_1$, soThe separable-profile closure for an age-structured population has a qualification: arbitrary age-dependent hazard functions do not give this constant-coefficient ordinary differential equation for arbitrary initial conditions. Without fixed $f_i$, the effective exit rates are $\int {\Gamma }_i{n}_{i}d\tau /N_i$ and can vary with time. Moreover, an exact positive separation of variables with both profiles fixed forces $N_2/N_1$ to be constant through the incoming fluxes, hence selects a common exponential mode. The displayed ordinary differential equation is the necessary integrated system for such profiles; it is also exact for arbitrary populations when both hazard functions are age independent.